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Absolute Hodge classes

Posted by Martin Orr on Thursday, 20 November 2014 at 18:55

Let A be an abelian variety over a field k of characteristic zero. For each embedding \sigma \colon k \hookrightarrow \mathbb{C}, we get a complex abelian variety A^\sigma by applying \sigma to the coefficients of equations defining A.

Whenever an object attached to A is defined algebraically, we will get closely related objects for each A^\sigma. On the other hand, whenever we use complex analysis to define an object attached to A^\sigma, we should expect to get completely unrelated things for different \sigma (if k = \mathbb{C} then most field embeddings k \hookrightarrow \mathbb{C} are horribly discontinuous so will mess up anything analytic).

Hodge classes provide a special case: the definition of Hodge classes on A^\sigma as H^{2p}(A^\sigma, \mathbb{Z}) \cap H^{p,p}(A^\sigma) is analytic so we expect no relation between Hodge classes on different A^\sigma. But the Hodge conjecture says that every Hodge class in H^{2p}(A^\sigma, \mathbb{C}) is an algebraic cycle class, and this implies the associated cohomology class in H^{2p}(A^{\sigma'}, \mathbb{C}) is also a Hodge class for every \sigma' \colon k \hookrightarrow \mathbb{C}. (We will explain in the post why there is a natural semilinear isomorphism H^{2p}(A^\sigma, \mathbb{C}) \to H^{2p}(A^{\sigma'}, \mathbb{C}).)

Deligne had the idea that we could pick this out as a partial step on the way to the Hodge conjecture: he defined an absolute Hodge class to be a cohomology class such that its associated class on A^\sigma is a Hodge class for every \sigma and proved that every Hodge class on an abelian variety is an absolute Hodge class. It turns out that this is sufficient to obtain some of the consequences which would follow from the Hodge conjecture. In this post we will explain the definition of absolute Hodge classes.

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